Oct 4 Tangent Spaces

Geometric Definition

The definition of a tangent space is the space of all vectors tangent to a surface at some point.

Tangent Spaces of Images

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If you've heard of domain/range, an image is just like a range. It is the set of all c s.t. $f(x) = c$.

It turns out the span of the Jacobian is the tangent space to an image. In every direction $f_x$, the partial represents the instantaneous rate of change, so you can create a tangent line at that point using the partial. Therefore, in every direction, you have a tangent line. The tangent space is the span of all of these lines.
$$\text{Span}\left(\begin{bmatrix}\frac{\partial f}{\partial x_1} & \frac{\partial f}{\partial x_2} & ... & \frac{\partial f}{\partial x_n} \end{bmatrix}\right)$$
Special case:
$$f: \mathbb{R}^n \to \mathbb{R}$$
If this is the case, you can write $w = f(x,y,z) \to f(x,y,z) - w = 0$. In this case, it's the same type of problem as a level set.

Tangent Spaces to Graphs

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A graph is pretty similar to what we normally think of graphs, they equal
$$\Gamma(x) = \{\begin{bmatrix}x\\f(x)\end{bmatrix} \mid x \in \mathbb{R}^n\}$$
Think about graphing $y = x^2$, you graph the x and y points together:
$$\begin{bmatrix}0 \\ 0\end{bmatrix}, \begin{bmatrix}3 \\ 9\end{bmatrix}, \begin{bmatrix}-2 \\ 4\end{bmatrix} \dots$$
The tangent space equals the graph of the derivative, meaning it's $$\{\begin{bmatrix}x\\Df(p)(x)\end{bmatrix} \mid x \in \mathbb{R}^n\}$$
The Jacobian for $\begin{bmatrix}x^2\end{bmatrix}$ is very simple: $\begin{bmatrix}2x\end{bmatrix}$. Therefore, the graph of the derivative is $x$ and the linear approximation at that point, which is $2x$ times $x$ = $2x^2$. Why is this? Imagine we had an x-value: $3$, the Jacobian would be $[6]$. We're graphing $\begin{bmatrix}x \\ Jf(p)(x)\end{bmatrix}$ = $\begin{bmatrix}x \\ 6x\end{bmatrix}$. Now, this is what we actually get in the general case:
$$\begin{bmatrix}x \\ 2x^2\end{bmatrix} = \text{Span}\left(\begin{bmatrix}1 \\ 2x\end{bmatrix}\right)$$

Tangent Spaces to Level Curves

In the post about gradients, we talked about gradients were orthogonal to level curves. So, how do we combine our understanding of tangent spaces: the span of the derivative (Jacobian) and the span of the graph of the derivative.

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The tangent space of a level curve is the kernel of the derivative. This means $Df(p) \cdot v = 0$.
If that's the case, then $Df(p)^{T}v = 0 \to \nabla f \cdot v = 0$ Therefore, $\nabla f$ is the normal vector to the tangent space.

Example

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Find the tangent space to $x^2 + y^2 - z^2 = 0$ at the point $(3,4,5)$.
The gradient is $\begin{bmatrix}2x\\2y\\-2z\end{bmatrix} = \begin{bmatrix}6\\8\\-10\end{bmatrix}$
The kernel of that means $\begin{bmatrix}6\\8\\-10\end{bmatrix}\cdot v = 0$, so the tangent space is just a plane with that normal vector.
$$6x + 8y - 10z = 0$$